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A flownet is a graphical representation of
two-dimensional steady-state groundwater flow
through aquifers.
Construction of a flownet is often used for solving groundwater
flow problems where the geometry makes analytical solutions
impractical. The method is often used in civil
engineering, hydrogeology or soil mechanics as a first check for
problems of flow under hydraulic structures like dams or sheet pile
walls. As such, a grid obtained by drawing a series of
equipotential lines is called a flownet. The flownet is an
important tool in analysing two-dimensional irrotational flow
problems.

Contents

Basic
method

The method consists of filling the flow area with stream and
equipotential lines, which are everywhere perpendicular to each other, making a curvilinear grid. Typically there are two
surfaces (boundaries) which are at constant values of potential or
hydraulic head (upstream and downstream ends), and the other
surfaces are no-flow boundaries (i.e., impermeable; for example the
bottom of the dam and the top of an impermeable bedrock layer),
which define the sides of the outermost streamtubes (see figure 1
for a stereotypical flownet example).

Mathematically, the process of constructing a flownet consists
of contouring the
two harmonic or analytic functions of potential and
stream
function. These functions both satisfy the Laplace equation and the contour lines
represent lines of constant head (equipotentials) and lines tangent
to flowpaths (streamlines). Together, the potential function and
the stream function form the complex potential, where the potential
is the real part, and the stream function is the imaginary
part.

The construction of a flownet only provides an approximate
solution to the flow problem, but it can be quite good even for
problems with complex geometries by following a few simple rules
(initially developed by Philipp Forchheimer around 1900,
and later formalized by Arthur Casagrande in 1937) and a
little practice:

streamlines and equipotentials meet at right angles (including the
boundaries),

diagonals drawn between the cornerpoints of a flownet will meet
each other at right
angles (useful when near singularities),

streamtubes and drops in equipotential can be halved and it
should still make squares (useful when squares get very large at
the ends),

flownets often have areas which consist of nearly parallel lines, which produce true
squares; start in these areas — working towards areas with complex
geometry,

many problems have some symmetry (e.g., radial flow to a well); only a section of the flownet needs
to be constructed,

the sizes of the squares should change gradually; transitions
are smooth and the curved paths should be roughly elliptical or parabolic in shape.

(It is often held that a circle inscribed in any square
should touch each side only once. It is worth pointing out that
this notion is wrong because it appears so often. It can be
disproved by examining the flownet associated with a single
well.)

Example
flownets

The first flownet pictured here (modified from Craig, 1997)
illustrates and quantifies the flow which occurs under the dam (flow is assumed to be invariant
along the axis of the dam — valid near the middle of the dam); from
the pool behind the dam (on the right) to the tailwater downstream from the dam (on the
left).

There are 16 green equipotential lines (15 equal drops in
hydraulic head) between the 5 m upstream head to the 1m downstream
head (4 m / 15 head drops = 0.267 m head drop between each green
line). The blue streamlines (equal changes in the streamfunction
between the two no-flow boundaries) show the flowpath taken by
water as it moves through the system; the streamlines are
everywhere tangent to the flow velocity.

Example flownet 2, click to view fullsize.

The second flownet pictured here (modified from Ferris, et al.,
1962) shows a flownet being used to analyze map-view flow
(invariant in the vertical direction), rather than a cross-section.
Note that this problem has symmetry, and only the left or right
portions of it needed to have been done. To create a flownet to a
point sink (a singularity), there must be a recharge boundary
nearby to provide water and allow a steady-state flowfield to
develop.

Flownet
results

Darcy's law
describes the flow of water through the flownet. Since the head
drops are uniform by construction, the gradient is inversely
proportional to the size of the blocks. Big blocks mean there is a
low gradient, and therefore low discharge (hydraulic conductivity
is assumed constant here).

An equivalent amount of flow is passing through each streamtube
(defined by two adjacent blue lines in diagram), therefore narrow
streamtubes are located where there is more flow. The smallest
squares in a flownet are located at points where the flow is
concentrated (in this diagram they are near the tip of the cutoff
wall, used to reduce dam underflow), and high flow at the land
surface is often what the civil engineer is trying to avoid, being
concerned about soil piping or
dam failure.

Singularities

Irregular points (also called singularities) in the flow
field occur when streamlines have kinks in them (the derivative doesn't exist
at a point). This can happen where the bend is outward (e.g., the
bottom of the cutoff wall in the figure above), and there is
infinite flux at a point, or where the bend is inward (e.g., the
corner just above and to the left of the cutoff wall in the figure
above) where the flux is zero.

The second flownet illustrates a well, which is typically represented
mathematically as a point source (the well shrinks to zero radius);
this is a singularity because the flow is converging to a point, at
that point the Laplace equation is not satisfied.

These points are mathematical artifacts of the equation used to
solve the real-world problem, and do not actually mean that there
is infinite or no flux at points in the subsurface. These types of
points often do make other types of solutions (especially numeric)
to these problems difficult, while the simple graphical technique
handles them nicely.

Extensions to standard
flownets

Typically flownets are constructed for homogeneous, isotropic porous media experiencing
saturated flow to known boundaries. There are extensions to the
basic method to allow some of these other cases to be solved:

anisotropic aquifer: drawing the flownet in
a transformed domain, then scaling the results differently in the
principle hydraulic conductivity directions, to return the
solution

one boundary is a seepage face: iteratively solving for both
the boundary condition and the solution throughout the domain

Although the method is commonly used for these types of
groundwater flow problems, it can be used for any problem which is
described by the Laplace equation
(),
for example electrical
current flow through the earth.